On a (very) tangential note, I attempted to get tickets for the Smithsonian Air & Space Museum in March shortly after it reopened by logging into their website promptly at 9a and the entire allotment was gone within 3 minutes. When we got down to the Mall, the line for people who had tickets was a solid 500 meters long and scalpers were asking $50/ticket, which would have simply earned us a chance to stand in the line for several hours. As much as I love airplanes, checking out the other museums seemed to be the logical course of action.
Switching gears back to Beijing, if I ever got shut out of the Forbidden City, I'd simply start off in Jingshan (to grab that choice photo against the Forbidden City in the background) and then Beihai instead, and maybe use the money saved on tickets to rent a boat. And, of course, I'd cap off the day with cold beer.